One piece not listed in your description is the fallback value of {Shoot Days}
, which appears to be the result if the number of shoot days is below 4 or above 15.
This can be simplified by only doing one comparison for each level. As long as you do them in order, it all works. This can be done low to high:
IF({Shoot Days} <= 4, {Shoot Days}, IF({Shoot Days} <= 8, 4, IF({Shoot Days} <= 15, 8, {Shoot Days})))
…or high to low:
IF({Shoot Days} >= 15, {Shoot Days}, IF({Shoot Days} > 8, 8, IF({Shoot Days} > 4, 4, {Shoot Days})))

thank you. I knew I was complicating it.